Video: Deducing the Velocity Profile of a Moving Body in a Viscous Fluid

Which of the graphs (a), (b), (c) and (d) most correctly shows how the velocity of an object changes with time if the object is subject to a constant force while moving through a fluid that exerts a drag force on it, bringing it to its terminal velocity? [A] Graph (a) [B] Graph (b) [C] Graph (c) [D] Graph (d)


Video Transcript

Which of the graphs a, b, c and d most correctly shows how the velocity of an object changes with time if the object is subject to a constant force while moving through a fluid that exerts a drag force on it, bringing it to its terminal velocity?

Okay, as we get started here, we’re told that we have some kind of object. And let’s say that this here is our object. We’re told further that our object is subject to a constant force. We can sketch that in on our object, say acting to the right. And we’ll simply call that force 𝐹. We don’t know what it is, but it’s just some force. In addition to this, we know that our object is moving through a fluid. A fluid recall is a liquid or a gas. And as a result, as the object moves, it experiences a drag force. That’s a force resisting its motion. Lastly, we know that the effect of this drag force is such that eventually our object is brought to its terminal velocity. We want to know which of these four graphs — a, b, c and d — most correctly represents how the velocity of our object changes in time.

To begin figuring that out, let’s clear a bit of working space on screen. Okay, so here again is our object, and it’s being acted on by this constant force we’ve called 𝐹. We know our object is moving through a fluid. And what if we just say that that fluid is water, that our object is moving through a tank of water? Well, in that case, we know what will happen as our object starts to move under the influence of the force 𝐹. As it begins to move to the right, there will be a drag force that emerges acting in the opposite direction. We can call it 𝐹 sub 𝐷 that opposes this motion.

And there’s an important difference between the drag force and this force we’ve called 𝐹. Recall that 𝐹 is a constant force. It never changes, whereas the drag force does change as the velocity of our object changes. In particular, the drag force will increase the faster our object moves. Now, before we get too far ahead of ourselves. Let’s notice that in all four cases for all four of our answer options, the initial velocity of our object is said to be zero.

So let’s do this. Let’s let our object in this tank of water start from rest. Beginning at that point, our object speed will clearly be zero. And if it has no speed, then there’s no force resisting its motion, since it has no motion. At the outset then, at the initial moment, we can say that there is no drag force on our object. The only force acting on it is this constant force 𝐹. Under the influence of that force, though, because it is a net force acting on our object, our object starts to accelerate. And the instant our object speeds up, the moment that it has a speed above zero, a small drag force appears opposing that motion. This drag force is caused by the interaction between this object and the water in the tank.

So our object’s velocity started at zero. And it’s increasing thanks to the fact that there are unbalanced forces acting on the object. And therefore, it accelerates. But here is where we want to recall that the drag force, unlike our constant force 𝐹, doesn’t stay the same. It’s generally the case that drag force increases as object velocity increases. And indeed, thanks to the fact that there’s an overall or net force on our object to the right, its velocity will increase. But then, as its velocity increases, so does the drag force. We see, though, that at this point the drag force is still less than the constant force 𝐹. So our object continues to speed up. And as it does so, the drag force grows and grows.

So long as these forces — our drag force and our constant force — are unbalanced, our object will continue to speed up. But the more it does so, the greater the drag force until eventually the two forces are equal. At this point, we can recall that when the net force on a projectile such as our object is zero, that means that object has reached what’s called terminal velocity. This is the point at which a projectile is no longer speeding up, but it’s reached its maximum speed. Because an object’s terminal velocity is its maximum speed, that means we can eliminate a couple of our possible answer options.

We see both for graph a as well as for graph d that the far-right portion of the graph in each case does not indicate a maximum velocity. In the case of graph a, that’s because we’ve already achieved a higher velocity earlier on. So graph a doesn’t correctly show an object reaching terminal velocity. Then, in the case of graph d, our velocity is still increasing as we get to the right most portion of the curve. In this case also, we’re not at a maximum value. We’re not at terminal velocity. If we look then at graph c, this graph shows us an object that is speeding up initially at an increasing rate. In other words, not only was the object getting faster, but the rate at which it got faster was growing.

But in the case of our object, acted on by the constant force 𝐹 and opposed by the drag force, what we saw instead was that while the object’s velocity did increase, it was increasing at a decreasing rate. That is, as the object’s velocity approached its maximum value, its velocity changed by smaller and smaller amounts. So it’s the shape of this part of the curve of graph c, which tells us that this is not a correct representation of our object’s velocity versus time. Yes, at first, our object was speeding up. But the rate at which it was speeding up was not increasing. It was decreasing. So option c can’t be our choice either.

When we look at the graph for option b, we see that this graph has the right shape to it. Velocity is increasing but increasing at a decreasing or smaller rate over time. And eventually, the velocity does level out to a set value, the object’s terminal velocity. It’s graph b then that correctly represents the relationship we’re looking for.

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